3.156 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(a g+b g x)^2 (c i+d i x)^3} \, dx\)

Optimal. Leaf size=381 \[ -\frac{b^3 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac{3 b^2 d \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (b c-a d)^4}+\frac{3 b d^2 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac{d^3 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac{b^3 B n (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}+\frac{3 b^2 B d n \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^2 i^3 (b c-a d)^4}-\frac{3 b B d^2 n (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4}+\frac{B d^3 n (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4} \]

[Out]

(B*d^3*n*(a + b*x)^2)/(4*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) - (3*b*B*d^2*n*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(
c + d*x)) - (b^3*B*n*(c + d*x))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (d^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(
c + d*x))^n]))/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) + (3*b*d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*
g^2*i^3*(a + b*x)) - (3*b^2*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^4*
g^2*i^3) + (3*b^2*B*d*n*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^4*g^2*i^3)

________________________________________________________________________________________

Rubi [C]  time = 1.08323, antiderivative size = 657, normalized size of antiderivative = 1.72, number of steps used = 30, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{3 b^2 B d n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac{3 b^2 d \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac{b^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac{3 b^2 d \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac{2 b d \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^3}-\frac{d \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^2}-\frac{b^2 B n}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac{3 b^2 B d n \log ^2(a+b x)}{2 g^2 i^3 (b c-a d)^4}+\frac{3 b^2 B d n \log ^2(c+d x)}{2 g^2 i^3 (b c-a d)^4}+\frac{3 b^2 B d n \log (a+b x)}{2 g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d n \log (c+d x)}{2 g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}+\frac{5 b B d n}{2 g^2 i^3 (c+d x) (b c-a d)^3}+\frac{B d n}{4 g^2 i^3 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^2*(c*i + d*i*x)^3),x]

[Out]

-((b^2*B*n)/((b*c - a*d)^3*g^2*i^3*(a + b*x))) + (B*d*n)/(4*(b*c - a*d)^2*g^2*i^3*(c + d*x)^2) + (5*b*B*d*n)/(
2*(b*c - a*d)^3*g^2*i^3*(c + d*x)) + (3*b^2*B*d*n*Log[a + b*x])/(2*(b*c - a*d)^4*g^2*i^3) + (3*b^2*B*d*n*Log[a
+ b*x]^2)/(2*(b*c - a*d)^4*g^2*i^3) - (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^2*i^3*(a
+ b*x)) - (d*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^2*g^2*i^3*(c + d*x)^2) - (2*b*d*(A + B*Log
[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^2*i^3*(c + d*x)) - (3*b^2*d*Log[a + b*x]*(A + B*Log[e*((a + b*x
)/(c + d*x))^n]))/((b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*n*Log[c + d*x])/(2*(b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*
n*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^4*g^2*i^3) + (3*b^2*d*(A + B*Log[e*((a + b*x)/(
c + d*x))^n])*Log[c + d*x])/((b*c - a*d)^4*g^2*i^3) + (3*b^2*B*d*n*Log[c + d*x]^2)/(2*(b*c - a*d)^4*g^2*i^3) -
(3*b^2*B*d*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*n*PolyLog[2, -
((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*n*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((
b*c - a*d)^4*g^2*i^3)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(156 c+156 d x)^3 (a g+b g x)^2} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)^2}-\frac{b^3 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^2 g^2 (c+d x)^3}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)^2}+\frac{b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2 (c+d x)}\right ) \, dx\\ &=-\frac{\left (b^3 d\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac{b^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3796416 (b c-a d)^3 g^2}+\frac{\left (b d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{1898208 (b c-a d)^3 g^2}+\frac{d^2 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3796416 (b c-a d)^2 g^2}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac{b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}+\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}-\frac{\left (b^2 B d n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3796416 (b c-a d)^3 g^2}+\frac{(b B d n) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1898208 (b c-a d)^3 g^2}+\frac{(B d n) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{7592832 (b c-a d)^2 g^2}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac{b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}+\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{1265472 (b c-a d)^4 g^2}-\frac{\left (b^2 B d n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3796416 (b c-a d)^2 g^2}+\frac{(b B d n) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1898208 (b c-a d)^2 g^2}+\frac{(B d n) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{7592832 (b c-a d) g^2}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac{b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}+\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^3 B d n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}-\frac{\left (b^3 B d n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}-\frac{\left (b^2 B d^2 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d^2 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3796416 (b c-a d)^2 g^2}+\frac{(b B d n) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1898208 (b c-a d)^2 g^2}+\frac{(B d n) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7592832 (b c-a d) g^2}\\ &=-\frac{b^2 B n}{3796416 (b c-a d)^3 g^2 (a+b x)}+\frac{B d n}{15185664 (b c-a d)^2 g^2 (c+d x)^2}+\frac{5 b B d n}{7592832 (b c-a d)^3 g^2 (c+d x)}+\frac{b^2 B d n \log (a+b x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac{b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}-\frac{b^2 B d n \log (c+d x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 B d n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}-\frac{b^2 B d n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^3 B d n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d^2 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1265472 (b c-a d)^4 g^2}\\ &=-\frac{b^2 B n}{3796416 (b c-a d)^3 g^2 (a+b x)}+\frac{B d n}{15185664 (b c-a d)^2 g^2 (c+d x)^2}+\frac{5 b B d n}{7592832 (b c-a d)^3 g^2 (c+d x)}+\frac{b^2 B d n \log (a+b x)}{2530944 (b c-a d)^4 g^2}+\frac{b^2 B d n \log ^2(a+b x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac{b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}-\frac{b^2 B d n \log (c+d x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 B d n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{b^2 B d n \log ^2(c+d x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 B d n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1265472 (b c-a d)^4 g^2}+\frac{\left (b^2 B d n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1265472 (b c-a d)^4 g^2}\\ &=-\frac{b^2 B n}{3796416 (b c-a d)^3 g^2 (a+b x)}+\frac{B d n}{15185664 (b c-a d)^2 g^2 (c+d x)^2}+\frac{5 b B d n}{7592832 (b c-a d)^3 g^2 (c+d x)}+\frac{b^2 B d n \log (a+b x)}{2530944 (b c-a d)^4 g^2}+\frac{b^2 B d n \log ^2(a+b x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3796416 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7592832 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1898208 (b c-a d)^3 g^2 (c+d x)}-\frac{b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1265472 (b c-a d)^4 g^2}-\frac{b^2 B d n \log (c+d x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 B d n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1265472 (b c-a d)^4 g^2}+\frac{b^2 B d n \log ^2(c+d x)}{2530944 (b c-a d)^4 g^2}-\frac{b^2 B d n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}-\frac{b^2 B d n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}-\frac{b^2 B d n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{1265472 (b c-a d)^4 g^2}\\ \end{align*}

Mathematica [C]  time = 0.806644, size = 477, normalized size = 1.25 \[ \frac{6 b^2 B d n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-6 b^2 B d n \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-12 b^2 d \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{4 b^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+12 b^2 d \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{8 b d (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-\frac{2 d (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}-\frac{4 b^3 B c n}{a+b x}+\frac{4 a b^2 B d n}{a+b x}+6 b^2 B d n \log (a+b x)-\frac{8 a b B d^2 n}{c+d x}+\frac{2 b B d n (b c-a d)}{c+d x}+\frac{B d n (b c-a d)^2}{(c+d x)^2}+\frac{8 b^2 B c d n}{c+d x}-6 b^2 B d n \log (c+d x)}{4 g^2 i^3 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^2*(c*i + d*i*x)^3),x]

[Out]

((-4*b^3*B*c*n)/(a + b*x) + (4*a*b^2*B*d*n)/(a + b*x) + (B*d*(b*c - a*d)^2*n)/(c + d*x)^2 + (8*b^2*B*c*d*n)/(c
+ d*x) - (8*a*b*B*d^2*n)/(c + d*x) + (2*b*B*d*(b*c - a*d)*n)/(c + d*x) + 6*b^2*B*d*n*Log[a + b*x] - (4*b^2*(b
*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (2*d*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]))/(c + d*x)^2 - (8*b*d*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) - 12*b^2*d*Log[a
+ b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*b^2*B*d*n*Log[c + d*x] + 12*b^2*d*(A + B*Log[e*((a + b*x)/(
c + d*x))^n])*Log[c + d*x] + 6*b^2*B*d*n*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*P
olyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 6*b^2*B*d*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*L
og[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/(4*(b*c - a*d)^4*g^2*i^3)

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Maple [F]  time = 0.716, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{2} \left ( dix+ci \right ) ^{3}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x)

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Maxima [B]  time = 1.83472, size = 2327, normalized size = 6.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

-1/2*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^3*
c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*
b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*g
^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^4
*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c^4
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n
) - 1/4*(4*b^3*c^3 - 15*a*b^2*c^2*d + 12*a^2*b*c*d^2 - a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 6*(b^3*d^3*x^
3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(b^3*d^3*x
^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(d*x + c)^2 - 3*(b^3*c^2*
d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d
+ 2*a*b^2*c*d^2)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d
+ 2*a*b^2*c*d^2)*x + 2*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2
)*x)*log(b*x + a))*log(d*x + c))*B*n/(a*b^4*c^6*g^2*i^3 - 4*a^2*b^3*c^5*d*g^2*i^3 + 6*a^3*b^2*c^4*d^2*g^2*i^3
- 4*a^4*b*c^3*d^3*g^2*i^3 + a^5*c^2*d^4*g^2*i^3 + (b^5*c^4*d^2*g^2*i^3 - 4*a*b^4*c^3*d^3*g^2*i^3 + 6*a^2*b^3*c
^2*d^4*g^2*i^3 - 4*a^3*b^2*c*d^5*g^2*i^3 + a^4*b*d^6*g^2*i^3)*x^3 + (2*b^5*c^5*d*g^2*i^3 - 7*a*b^4*c^4*d^2*g^2
*i^3 + 8*a^2*b^3*c^3*d^3*g^2*i^3 - 2*a^3*b^2*c^2*d^4*g^2*i^3 - 2*a^4*b*c*d^5*g^2*i^3 + a^5*d^6*g^2*i^3)*x^2 +
(b^5*c^6*g^2*i^3 - 2*a*b^4*c^5*d*g^2*i^3 - 2*a^2*b^3*c^4*d^2*g^2*i^3 + 8*a^3*b^2*c^3*d^3*g^2*i^3 - 7*a^4*b*c^2
*d^4*g^2*i^3 + 2*a^5*c*d^5*g^2*i^3)*x) - 1/2*A*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*
d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*
a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^
3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^
3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^
2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3
))

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Fricas [B]  time = 0.582539, size = 1956, normalized size = 5.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/4*(4*A*b^3*c^3 + 6*A*a*b^2*c^2*d - 12*A*a^2*b*c*d^2 + 2*A*a^3*d^3 + 6*(2*A*b^3*c*d^2 - 2*A*a*b^2*d^3 - (B*b
^3*c*d^2 - B*a*b^2*d^3)*n)*x^2 + 6*(B*b^3*d^3*n*x^3 + B*a*b^2*c^2*d*n + (2*B*b^3*c*d^2 + B*a*b^2*d^3)*n*x^2 +
(B*b^3*c^2*d + 2*B*a*b^2*c*d^2)*n*x)*log((b*x + a)/(d*x + c))^2 + (4*B*b^3*c^3 - 15*B*a*b^2*c^2*d + 12*B*a^2*b
*c*d^2 - B*a^3*d^3)*n + 3*(6*A*b^3*c^2*d - 4*A*a*b^2*c*d^2 - 2*A*a^2*b*d^3 - (B*b^3*c^2*d + 2*B*a*b^2*c*d^2 -
3*B*a^2*b*d^3)*n)*x + 2*(2*B*b^3*c^3 + 3*B*a*b^2*c^2*d - 6*B*a^2*b*c*d^2 + B*a^3*d^3 + 6*(B*b^3*c*d^2 - B*a*b^
2*d^3)*x^2 + 3*(3*B*b^3*c^2*d - 2*B*a*b^2*c*d^2 - B*a^2*b*d^3)*x + 6*(B*b^3*d^3*x^3 + B*a*b^2*c^2*d + (2*B*b^3
*c*d^2 + B*a*b^2*d^3)*x^2 + (B*b^3*c^2*d + 2*B*a*b^2*c*d^2)*x)*log((b*x + a)/(d*x + c)))*log(e) + 2*(6*A*a*b^2
*c^2*d - 3*(B*b^3*d^3*n - 2*A*b^3*d^3)*x^3 - 3*(3*B*a*b^2*d^3*n - 4*A*b^3*c*d^2 - 2*A*a*b^2*d^3)*x^2 + (2*B*b^
3*c^3 - 6*B*a^2*b*c*d^2 + B*a^3*d^3)*n + 3*(2*A*b^3*c^2*d + 4*A*a*b^2*c*d^2 + (2*B*b^3*c^2*d - 4*B*a*b^2*c*d^2
- B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)))/((b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^
2*c*d^5 + a^4*b*d^6)*g^2*i^3*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*
a^4*b*c*d^5 + a^5*d^6)*g^2*i^3*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*
b*c^2*d^4 + 2*a^5*c*d^5)*g^2*i^3*x + (a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*
c^2*d^4)*g^2*i^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**2/(d*i*x+c*i)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (b g x + a g\right )}^{2}{\left (d i x + c i\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)/((b*g*x + a*g)^2*(d*i*x + c*i)^3), x)